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HW3_2005

# Digital Signal Processing: A Practical Approach (2nd Edition)

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C/D y 1 [ n ] x 1 ( t ) x 2 ( t ) T =12kHz C/D C H ( z ) x 1 ( t ) x 2 ( t ) T 1 =12KHz C/D A T 2 =6KHz B y 2 [ n ] 2 2 x [ n ] y 1 [ n ] H 1 ( e j ω ) H 2 ( e j ω ) x [ n ] Homework Set #3: Continuous to Discrete Time Translation 1. The signal x c ( t ) = 2cos(200 π t ) is sampled with sampling periods, (a) T = 1/300; (b) T = 1/150; (c) T = 1/100; (d) T = 1/50 seconds, what are the resulting sampling sequences, x [ n ]? 2. The signal x c ( t ) = sin(200 π t ) + cos(50 π t ) is sampled with sampling periods, (a) T = 1/500; (b) T = 1/200; (c) T = 1/100; (d) T = 1/50 seconds, what are the resulting sampling sequences, x [ n ]? 3. Consider the system show below. For each of the following input signal x [ n ], indicate whether the output x r [ n ] = x [ n ]: (a) x [ n ] = cos( π n /5); (b) x [ n ] = cos( π n /4); (c) x [ n ] = cos( π n /3); (d) x [ n ] = n cos( π n /2); (e) x [ n ] = sin( π n ) 4. Consider audio signal sampling and reconstruction, we plan to design 6-time oversampling system (by using ideal digital low pass filter) to reduce the burden of the design of analog filters. The original signal sampling and reconstruction systems are shown as: (a) Specify the original frequency responses of H a ( j
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